extra storage for MacBook Air (11-inch, 2013),

That’s easy and you have two choices. My first MBAir didn’t have enough space for my music either. I purchased an SD Card (I provided an example link) and put my music on it. Writing data to an SD card isn’t terribly fast but the computer can read if plenty quickly for music and video. You might also be interested in a bus powered portable SDD (again, I’ve provided an example link.) I’m not specifically recommending those items, just giving you an idea of what to look for. The advantage of the SD Card is that you can keep it plugged in all the time. It will extend out from the SD port by about 1/4 inch. I safely kept mine in for 4 years without a problem. The advantage of the SSD is speed of writing and price per GB.

As to how to do it. See this article: https://www.imore.com/how-move-your-itunes-library-external-hard-drive

Just be sure that the drive is plugged in before you start the Music app and all will be well.

The MBAir is not upgradeable. If you are running out of space you'll have to figure out what can be safely offloaded to an external hard drive.